Behaviour of tapered piles subjected to lateral harmonic loading

  • Sonal Singh
  • Nihar Ranjan PatraEmail author
Technical Note


In this study, the behaviour of conventional concrete tapered piles subjected to lateral harmonic excitation is assessed. For a constant volume of pile material, the tapered pile is modelled in two different series. The first series is modelled with a constant pile tip diameter and the second series comprises of constant length of the pile. The response curves of the soil–pile system under horizontal vibration are obtained from three-dimensional finite element numerical simulation. The dynamic properties of the soil–pile system, i.e., the stiffness and the damping ratio are calculated from the response curves. It is observed that the tapered piles in the first series exhibit 4.14–8.87% lower resonant frequency than cylindrical piles of the same volume of pile material, whereas the tapered piles in the second series display a resonant frequency 2.95–7.69% higher than that of the cylindrical piles. Further, the resonant amplitude and frequency obtained from the finite element simulation are compared with a modified analytical solution. The resonant frequencies obtained from the finite element simulation are within an error limit of ± 0.57–10% as compared to that obtained from the analytical approach.


Dynamic response Tapered pile Finite element method Resonant frequency Stiffness Damping 

List of symbols


Parameters defined for calculation of damping ratio


Amplitude of the mth cycle

\(A_{m + 1}\)

Amplitude of the (m + 1)th cycle


Static displacement


Resonant amplitude used in the graphical method


Acceleration in m/s2


Horizontal displacement in m

\(\left( {A_{x} } \right)_{{\mathrm{max}}}\)

Amplitude at resonance


Damping coefficient of the pile


Embedment depth of the pile cap


Thickness of the pile cap


Eccentricity of the masses

\(E_{\text{p}} I_{\text{p}}\)

Bending stiffness of the pile segment j


Circular operating frequency in Hz


Restoring force of the pile


Acceleration due to gravity


Shear modulus of the soil adjacent to the segment j


Height of the element j

\(h_{n}^{\prime }\)

Non-dimensional parameter defined for calculation of soil resistance


Stiffness of the soil–pile system


Mass of the pile per unit length


Bending moment of the pile

\(M_{j1} ,M_{j2}\)

Bending moment at node 1 and 2 of element j, respectively


Mode number


Static load acting on the pile

\(p\left( z \right)\)

Soil reaction due to horizontal pile displacement \(u\left( z \right)\)


Applied harmonic load

\(u\left( z \right)\)

Horizontal pile displacement

\(u_{j1} ,u_{j2}\)

Displacement at node 1 and 2 of element j, respectively


Modal amplitude independent of \(z\)


Shear force

\(V_{j1} ,V_{j2}\)

Shear force at node 1 and 2 of element j, respectively


Weight of the eccentric masses


Depth along the pile length

\(\theta_{j1}\), \(\theta_{j2}\)

Rotation at node 1 and 2 of element j, respectively


Damped natural frequency


Damping ratio of the soil–pile system


Undamped natural frequency


Magnification factor


Angular excitation frequency (rad/s)


Logarithmic decrement


Intersection of the natural frequency curve with x-axis


Rotation amplitude of the pile

\((\alpha_{h} )_{1} , (\alpha_{h} )_{2}\)

Real and imaginary part of the soil reaction, respectively


Resonant frequency used in graphical method


Circular excitation frequency


Soil resistance factor


Soil resistance factor in nth mode


Natural frequency


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringIndian Institute of TechnologyKanpurIndia

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